Cosmological Implications of X-ray Clusters of Galaxies

Cosmological implications of clusters of galaxies are discussed with particular attention to their importance in probing the cosmological parameters. More specifically we compute the number counts of clusters of galaxies, Log $N$ -- Log $S$ relation, in X-ray and submm bands on the basis of the Press--Schechter theory. We pay particular attention to a set of theoretical models which well reproduce the {\it ROSAT} 0.5-2 keV band Log $N$ -- Log $S$, and explore possibilities to break the degeneracy among the viable cosmological models.